Pictures would help.
But if it's like the hollow version of the harmonica bridge, there is a little half round cottlier pin (However you spell it.) that holds the screw in place on one side of the bridge. Pop that out with something thin and pointy, and the screw should easily screw out.
Interesting... It almost appears in the photos that the end of the screw is peened (mushroomed) over after it is fully inserted into the body of the bridge. The only way to remove it if that is the case is to deform or removed the mushroomed part of the end, install the new saddle, and then peen it over again to retain it. This procedure is doable if you have some basic metal working skills and tools, but if you have never done it before it may be difficult to do and have it look "good as new". A "newbie" could probably make it functional though. Clips or springs are the more common method of manufacturing these things.
The one place I have a hard time seeing in the photos is the inside of the frame work on the far side away from the screw heads. If there is no clip there, I dunno. Probably peened in.